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NUMERICAL EXPERIMENTS WITH A FIVE-LEVEL 

GLOBAL ATMOSPHERIC PREDICTION MODEL 

USING A STAGGERED, SPHERICAL, SIGMA 

COORDINATE SYSTEM 



William Theodore Eli as 



m 



Library 

Naval Postgraduate Scfiod 

Monterey, California 93940 



\S iJ t %^ 'l^ a t^ %3 SI i im W^lflWUla 

ORlepey, California 




1^ ii 







NUMERICAL 
GLOBAL 
USING A 


EXPERIMENTS 
ATMOSPHERIC 
STAGGERED , 
COORDINATE 


WITH A FIVE-LEVEL 
PREDICTION MODEL 
SPHERICAL, SIGMA 

SYSTEM 






by 










William Theo 


dore Elias 
George 


Haltiner 


Th 


esis Advisors: 


Roger Williams 1 



March 1973 



T 



■'^'? 



Appnjovzd. {^OJi pabtlc n.zt^ja^e,; dUt/uhvution urJianittd. 



Numerical Experiments with a Five-Level 

Global Atmospheric Prediction Model 

Using a Staggered, Spherical, Sigma 

Coordinate System 

by 

William Theodore Elias 

Lieutenant, United States Navy 

B.S., California Polytechnic College at Pomona, I968 



Submitted in partial fulfillment of the 
requirements for the degree of 



MASTER OP SCIENCE IN METEOROLOGY 

from the 

NAVAL POSTGRADUATE SCHOOL 
March 1973 



Library 

J^aval Postgraduate School 

'Monterey. California 93940 



ABSTRACT 



Three cases of analytic data and one case of real data 
were numerically integrated using a 5-level baroclinic 
primitive equations model of the general circulation. 
Experiments were performed using initial winds derived from 
the linear balance equation and also winds derived analyti- 
cally. The feasibility of using the linear balance 
equation to initialize the wind field v/as examined. In all 
cases, the forecasts remained meteorological and reasonably 
well-behaved. Nevertheless, the forecasts derived from 
initial winds generated by the linear balance equation 
excited large, operationally-undesirable inertial-gravity 
waves, while the forecasts from analytically determined 
Initial winds remained virtually free of such small scale 
"noise". 



TABLE OF CONTENTS 

I. INTRODUCTION 11 

II. BAROCLINIC PRIMITIVE EQUATION MODEL 12 

A. PRIMITIVE EQUATIONS 12 

B. GRID m 

C. VERTICAL LAYERING lH 

D. TIME DIFFERENCING 17 

III. INITIAL CONDITIONS 19 

A. ANALYTIC WINDS 22 

B. DERIVED WINDS 23 

IV. FOURIER WAVE ANALYSIS METHOD 25 

V. RESULTS 26 

VI. CONCLUSIONS 5.6 

LIST OF REFERENCES 58 

INITIAL DISTRIBUTION LIST 60 

FORM DD 1473 64 



LIST OF CHARTS 



A. Initial Surface Pressure Analysis 

(Experiment IV) ^0 

B. Verifying Surface Pressure Analysis 

(Experiment IV) ^1 

C. 12-Hour Forecast Using FNWC Model 

(Experiment IV) 42 

D. 12-Hour Forecast Using Global Model 

(Experiment IV) ^3 

E. Initial Surface Pressure Analysis 

(Experiment I) 44 

F. 12-Hour Surface Pressure Forecast Using Analytic 

and Derived Winds (Experiment I) 45 

G. 24-Hour Surface Pressure Forecast Using Analytic 

and Derived Winds (Experiment I) 46 

H. 36-Hour Surface Pressure Forecast Using Analytic 

and Derived Winds (Experiment I) .47 

I. Initial Surface Pressure Analysis 

(Experiment II) 48 

J. 12-Hour Surface Pressure Forecast Using Analytic 

and Derived Winds (Experiment II) 49 

K. 24-Hour Surface Pressure Forecast Using Analytic 

and Derived Winds (Experiment II) 50 

L. 36-Hour Surface Pressure Forecast Using Analytic 

and Derived Winds (Experiment II) 51 

M. Initial Surface Pressure Analysis 

(Experiment III) 52 

N. 6-Hour and 12-Hour Surface Pressure Forecast 

Using Analytic Winds (Experiment III) 53 

0. 18-Hour and 24-Hour Surface Pressure Forecast 

Using Analytic Winds (Experiment III) 54 

P. 30-Hour and 36-Hour Surface Pressure Forecast 

Using Analytic Winds (Experiment III) 55 



LIST OF FIGURES 

1. Location of Variables 15 

2. Vertical Layering l6 

3. Plot of Terrain Pressure Versus Forecast 

Hour at 0°N 15°E (Experiment IV) 3^ 

k. Plot of Terrain Pressure Versus Forecast 

Hour at hO°n 15° E (Experiment IV) 35 

5. Plot of Terrain Pressure Versus Forecast 

Hour at 85°N 15°E (Experiment IV) 36 

6. Plot of Terrain Pressure Versus Forecast 
Hour at 0°N 15°E for "Analytic" Winds 

(Experiment II) 37 

7. Plot of Terrain Pressure Versus Forecast 
Hour at 40°N 15°E for "Analytic" Winds 
(Experiment II) 38 

8. Plot of Terrain Pressure Versus Forecast 
Hour at 85°N 15°E for "Analytic" Winds 
(Experiment II) 39 



LIST OF TABLES 



I. Wave Amplitudes and Mean Heights For 

Selected Latitutdes and Forecast Time 31 

II. Computed Phase Speeds at Selected Latitudes 
Along with Method of Balancing and the Time 
Required To Make A 36-Hour Forecast For 

Each Experiment 32 

III. Comparison of 12-Hour Forecast Pressures of 
the FNWC Model and the Global Model Against 

the Verifying Pressure Analysis 33 



LIST OF SYMBOLS AND ABBREVIATIONS 

A - Arbitrary constant in the stream function 

A - Arbitrary constant for the Fourier series 

^ cosine terms 

a - Earth's radius 

B - Arbitrary constant in the stream function 

B - Arbitrary constants for the Fourier series 
m . , 

sine terms 

CDC - Control Data Corporation 

C - Arbitrary constants for the Fourier series 

^ combined terms 

Op - Specific heat for dry air at constant pressure 

c - Wave speed 

D - Lateral diffusion for the quantity indicated 
by the subscript 

D„ - Lateral diffusion of heat 

D - Lateral diffusion of specific humidity 

D - Lateral diffusion of momentum 
m 

FNWC - Fleet Numerical Weather Central 

F •• - Frictional stress 

f - Coriolis parameter 

f - Coriolis parameter at 45° north 

g - Acceleration of gravity 

H - Diabatic heating 

1 - Grid index in the x-direction (east-west) 

J - Grid index in the y-direction (north-south) 



M - Stands for <}), T, q, and w 

m - Wave number 

mb - Millibars 

NPS - Naval Postgraduate School 

NACA - National Advisory Committee on Aeronautics 

N - Wave num;ber plus one - m + 1 

n - Degree of the Legrendre function 

P - Pressure 

P ' - Legrendre function of order m and degree n 

Q - Moisture source/sink term 

q - Specific humidity 

R - Specific gas constant for dry air 

t - Time 

u - Zonal Wind 

^ - Horizontal vector velocity - (u,v) 

V - Meridional wind 

W - Stands for u and v 

w - Measure of vertical velocity, positive upward 
w = - g = - da 
dt 

At - Time increment 

Ax - Distance increment in the x-direction - a AX cos 

a - Specific volume 

6 - Phase angle for wave number m 
m ^ 

6 - Latitude, positive northward from south pole 

AG - Distance increment in the latitudinal direction 

X - Longitude, positive eastward from Greenwich 



8 



AX - Distance increment in longitudinal direction 

V - Angular v/ave velocity 

TT - Terrain pressure parameter 

a - Dimensionless vertical coordinate, <_ a £ 1, 
increasing downward 

• dcT 

G - Measure of vertical velocity - ^, -w 

<}) - Geopotential 

ip - Stream function 

V - Del operator (horizontal) 

2 

V - Laplacian operator (horizontal) 



ACKNOVJLEDGEKENTS 

The author wishes to express his thanks to Dr. G. J. 
Haltiner for his encouragement to undertake this project. 
Dr. R. T. Williams for his patient guidance without which 
this project would never have been completed, and Dr. P. 
J. Winninghoff for providing the original program. 

In addition, the computer support of FNWC, under the 
command of Captain W. S. Houston Jr. USN, was both essential 
and greatly appreciated. The author would also like to 
express his thanks to several members of Captain Houston's 
staff, particularly Commander Celia L. Barteau, USN, and 
Mr. Leo Clarke who provided coordination of computer 
services. 



10 



I. INTRODUCTION 

In the field of operational numerical weather prediction, 
the tendency, in I'ecent years, has been toward the develop- 
ment of sophisticated global prediction models. This has 
been made possible by the rapid growth of computing capacity 
and developments connected with general circulation research. 

The purpose of this study vias to examine a baroclinic 
primitive equation model using a global, staggered, spherical, 
Sigma coordinate system which could be used on an operational 
day to day basis by the United States Navy. In order to do 
this, an input procedure was developed to allow a real time 
Initialization from FNWC analysis fields. A set of analytic 
fields using an analytic spherical harm.onic stream function, 
first presented by Neamtan (19^6), was also developed for 
use as a controlled set of initial conditions. By using 
these analytic cases, errors in real data collection, 
analysis and initialization, which are inevitable in pract: - 
cal meteorology, were avoided. The use of an analytic case 
also allowed the control of temperature and moisture distri- 
bution, predominant wave number, phase speed and wave 
amplitude. 

The objective was to isolate and correct problems in 
the model by using the controlled analytic input with the 
ultimate aim of improving and extending the Navy's overall 
capability to predict weather phenomena on a global scale. 



11 



II. BAROCLINIC PRIMITIVE EQUATION MODEL 

The governing differential equations, written in vector 
form, are similar to sets used by Smagorinsky et al. (1965), 
Arakawa et al. (1969) and Kesel and Winninghoff (1972). 
The integrations were carried out on a global, spherical, 
staggered, sigma. coordinate system using the conservative- 
type difference equations based on the work of Arakawa (1966). 
The complete set of finite difference equations are given 
in Kesel and Winninghoff (1972) for the non-staggered 
square grid. The Arakawa type spatial differencing was used 
to eliminate the spurious energy growth which can occur 
with the more conventional finite difference approximations 
to the nonlinear advection terms. 

A. PRIMITIVE EQUATIONS 

The vector equation of horizontal motion in the sigma 
(a) coordinate system for this model is: 



^(ttV) + (V-7tV)V + 7T ^(aV) + f(KX7TV) + TrVcf) + QTraVTT = F + D„ 
d u do in 



The Thermodynamic energy equation is: 



iTr(^T) + V-(TrTV) + it |-(Ta) - — (a|^ + aV-VTT + -no) = ttH + Dn, 

oil do Cpdo J. 



12 



The mass continuity equation is: 



|i+ vc.^) + .||= 



The moisture continuity equation is : 



|^(7Tq) + V.(7TcV) + 7T l^-Cqa) = ttQ + D^ 



These equations are supplemented by the hydrostatic 
equation 



?^ + TTa = 
9a 



the equation of state 

a = RT/P 
and the dimensionless vertical coordinate 

a = P/tt 

In the above equations, u is the zonal wind component; 
Vj the meridional wind component; tt, the terrain pressure; 
a the vertical-velocity measure; T, the temperature; f, 
the Coriolis parameter; 6, the geopotential; q, the specific 
humidity; H, the diabatic heating; F, the frictional stress; 



13 



D , the lateral diffusion of momentum: D^, the lateral 

diffusion of heat; D , the lateral diffusion of moisture: 

' q ' 

Q, the moisture source/sink term and P, the pressure. 
Furthermore, the formulations for frictional stress (F), 
lateral diffusion (D), heating (H) and moisture (Q) are 
identical to those described in Kesel and Winninghoff (1972). 

B. GRID 

The spatial finite differencing was performed on a 
staggered, spherical grid. The geopotential ((})), tempera- 
ture (T), specific humidity (q) and the vertical-velocity 
measure (w = - a) were carried at the poles. The longitudinal 
and latitudinal grid increments were both five degrees. 
This gives 2520 points over the globe, with (4',T,q,w) 
and (u,v) carried at 1260 points each (See Fig. 1). When 
(j>,T,q or w variables are needed at (u,v) points (e.g. T 
in the friction term and w in the vertical advection of 
momentum term) , they are defined as the average of the 4 
values at the surrounding (4'>T,q,w) points. 

C . VERTICAL LAYERING 

The present model divides the atmosphere into five 
layers, as sketched in Figure 2. The basic variables of 
the model are carried at the center of each layer. Addi- 
tional variables and conditions are specified at the 
interface between layers, as well as the upper and lower 
boundaries . 



ih 



M (North Pole) 

W M W M .... M 
M V/ M W .... W 






W M W M .... M 

W 





M 


W 


M 


W 


+ 










J- 


W 


M 


W 


M 



• • • • 



M (South Pole) 



FIGURE 1. LOCATION OF VARIABLES 

M stands for 4), T, q and w variables 
W stands for u and v variables 



15 



M 



Level 
Computed Variables (Notational 

(at each level) Subscript) Sigma 
Pressure (a) 

Q Q w = UPPER BOUNDARY ^ q 



1 X IT 



.u^v^T^^ 5 . 



2 X 7T ^ ^ 0.2 



3 X IT 



.H-.Y-.T^^ i . 3 

.4 X TT "^ ^ 0.4 

.5 X TT H-LY^T^*-.q 3 0^5 

.6 X 7T "^ ? 0.6 

.7 X ^ HiY^T^^aS 2 Q^^ 

8 X TT "^ 1 ■ 0.8 

9 X TT "^xIjilA^S. I 0.9 

] 



w 


1 


^xijilA^s. 

w = 


1 

LOWER BOUNDARY 



FIGURE 2. VERTICAL LAYERING 

In the above figure, sigma (a) is the dimensionless 
vertical coordinate, u, is the zonal wind component, 
V is the meridional wind component, q is the specific 
humidity, <{) is the geopotential , it is the terrain , 
pressure and w is the vertical-velocity measure (-a). 



o 



16 



Integrations are performed in the Phillip's (1957) 
Sigma (a) coordinate system, where pressure, P, is normalized 
with the underlying terrain pressure, tt. The dimensionless 
vertical coordinate, sigma (a), is defined as 

a = P/tt 

It follows rhat 

0=1 at the earth's surface 
and a = at the top of the atmosphere. 

The zonal wind component (u), the meridional wind compo- 
nent (v) , the temperature (T) and geopotential ((j)) are 
carried at sigma levels .9> .7, .5, .3 and .1. The specific 
humidity (q) is carried at sigma levels .9, .7 and .5. The 
vertical velocity measure, w = -a, is calculated diagnosti- 
cally from the continuity equation for the layer interfaces. 
The vertical-velocity measure (w) is carried at sigma lev is 
.8, .6, .4 and .2. 

D. TIME DIFFERENCING 

The two finite difference techniques that were used to 
step forward in time are the centered (leapfrog) difference 
schem.e and the Matsuno (Euler backward) difference scheme. 
The finite difference equation for the centered technique is: 



pt+1 ^ pt-1 ^ 2At ^^' 



8t 



17 



Since the form has three time levels, it has both a physical 
and a computational mode (Haltiner, 1971). This computational 
mode, which changes phase at every time step, causes the solu- 
tions at adjacent time steps to become decoupled. Further- 
more, the centered scheme is not feasible for the first time 
step. Therefore, the Matsuno forv/ard time differencing scheme 
is used for the initial time step of each 6-hour integration 
cycle. This time step not only reduces solution separation 
but also selectively dampens high frequency waves (Haltiner, 
1971). The finite difference equation for the Matsuno scheme 
is: 

F* = F^ + At ^^ 



dt 



pt+l = pt ^ ^^ |Fi 

dt 



A time step of ten minutes was used for all experiments. 
This large time step is comiputationally stable when the 
Arakawa technique (Langlois and Kwok, I969) of averaging 
quantities involved in the longitudinal derivatives is use u 
Without this averaging technique, the von Neumann linear 
computational stability criterion (Haltiner, 1971) would 
require a 2.5 minute time step because the longitudinal grid 
distance at 85° north and south is only 47 kilometers. No 
averaging was done equatorward of 60° north and south. 

It should be noted at this time that lateral diffusion, 
friction, convective adjustment, convective condensation and 
large scale condensation are computed at every time step 

while heating is computed every 6 time steps (Kesel and 

Winninghoff, 1972). 

18 



III. INITIAL CONDITIONS 

Both real and analytic data were used for initialization. 
The data was produced external to the main program on a 
Northern Hemispheric FNWC 63 x 63 grid. The analytic data 
with known phase propagation properties was produced 
following the work of Neamtan (19^6), Gates (1962) and 
Heburn (1972). The real data was read from FNWC tapes. 
These fields were analyzed by the FNWC objective schemes. 

The main program was used to interpolate the data so 
that values at 5° latitude-longitude intersections were 
available over the Northern Hemisphere. The data was then 
reflected into the Southern Hemisphere. This procedure 
allows for an excellent test of the program. The model 
begins with fields which are mirror images of each other 
and they should remain so. In all cases, the input data 
consisted of the temperature analyses for the Northern 
Hemisphere at 12 constant pressure levels distributed from 
1000 to 50 mb, height analyses at 10 of these levels, 
moisture analyses at 4 levels from the surface to 500 mb, 
sea level pressure and sea surface temperature. Monthly 
mean surface temperature fields were used to derive an 
albedo field (Dickson and Posey, I967). The terrain height, 
in the real data case, was that used by the current FNWC 
model. The terrain height, in the analytic data cases, was 
set to zero. 



19 



In order to Isolate and evaluate certain numerical errors, 
it was decided to use analytic initial conditions for the 
stream function il^, from which the true solution of the 
complete vorticity equation in two dimensions could be 
obtained. With the assumption of nondivergent horizontal 
flow, harmonic wave solutions of the complete vorticity 
equation have been obtained for the sphere by Neamtan (19^6). 
These solutions yield the velocity of propagation of the 
waves. The solution for the ^ field was found to be 

i|; = A sin(mA-vt) P^^(sin G) - B r^ sin 6 + C Pj^(sln 6) 

(1) 

where A, B and C are constants to be determined, r is the 
radius of the earth, m is the hemispheric wave number, v/m 
is the angular phase speed of the wave disturbance 
(radians / second) , P denotes the Legendre polynomial and 
P ^ represents the associated Legendre function. 

If we choose C = (Gates, 1962) and n = m+1 [Buringt n 
and Torrance (1936), Kreyszig (1962)] equation (1) reduces to 

ip = A sin(mX-vt)(2N!/2\! ) sin (f) (cos(j)) - B r^ sin <|) . 

(2) 

The constant A is arbitrary and proportional to the wave 
amplitude. But it was shown by Haurwitz (19^0) that the 
solution obtained for i> implies the existence of a velocity 
distribution over the sphere such that the angular velocity 
of the westerly current is made up of the sum of three terms. 



20 



The first term, B, is constant over the sphere; the second 
term varies along a meridian but is constant round a circle 
of latitude; and the third term represents a harmonic wave 
which is propagated zonally with a constant angular velocity, 
v/m, which is given by the formula 



/ □ n(n+l) - 2 2fi f^. 

v/m = B — ^^ / , ., \ / . -, V (3) 

nCn+1) n(n+l) ^-^^ 



where fi is the earth's rotation speed. 

A reasonable meteorological pattern was obtained by 

2 —1 
choosing A as 1000 m sec , m as 6 and v/m as -13°, 0° and 

+12° longitude per day respectively. The ip field thus 

obtained was used to determine the initial fields. 

The geopotential fields were derived by solution of the 

linear balance equation 



V^ (j) = [f V^4; + Vi|;- A f] (^) 



over the entire Northern Hemisphere. Equation (4) was 
solved using the over-relaxation iterative technique with a 
relaxation tolerance of one meter (Haltiner, 1971) • 

The temperatures generated for the analytic cases were 
constant over each pressure surface and were consistent with 
the National Advisory Committee for Aeronautics (NACA) 
standard atmosphere (Haltiner and Martin, 1957). The eleva- 
tion 10 ,769 meters represents the tropopause in this 



21 



atmosphere. The following relations were used in calculating 
the analytic temperatures. 

Tp(<^K) = 288 - 0.0065 Z Z <_ 10,769 meters 

Tp(°K) =218 • Z 1 10,769 meters 

Zp(m) = i|4,308 [1 - (^^l_^)0-'-90233 ^ 1 10,769 meters 

Zp(m) = 10,769 + 6381.6 ln( ^^ p ^^ ) Z i 10,769 meters 

In the foregoing equations, the pressure is in millibars and 
the height Z, is in meters. In addition, the specific 
humidity (q) at the lowest three levels was set to zero and 
the forecast procedure excluded friction, convection adjust- 
ment, large scale convective condensation, large scale 
condensation, heating and terrain influences. 

The initial wind fields in the analytic cases were 
obtained in two ways. Winds obtained analytically from tl 3 
original ^ fields were called "analytic winds." Winds 
obtained from the linear balance equation were called 
"derived winds." It should be noted that "derived winds" 
v;ere used for the real data case. 

A. ANALYTIC WINDS 

The initial wind fields for the "analytic wind" cases 
were derived directly by differentiation of the initial 



22 



rl> field (Neamtan, 19^6). The formulas for the non-divergent 
wind components are 

u = - - (a sin(mX-vt)(-^^)[(cos 6)"^- 6 (sin e )^ (cos 6)^] 

- B r^ cos e I (5) 



V = r A ( . ' ) sin 6 (cos G) m cos(mX-vt) (6) 



where a is the earth's radius, A is an arbitrary constant 
that is proportional to the wave amplitude, 6 is the latitude, 
V is the angular velocity of the wave, A is the longitude, 
m is the wave number and N = m+1. 

B. DERIVED WINDS 

The "derived wind" field was obtained from a ^ field 
using the finite difference expressions 



1 ^\b 
^ = " i Ae 



and 



1 ^± 
a cos 6 AX 



where a is the earth's radius, AS is the latitudinal 
distance increment and AX is the longitudinal distance 
increment. The rp and 4) fields were obtained in two parts 



23 



depending on latitude. Poleward of 25° north and south , the 
initial* geopotential field, say <t> , was retained and the ^ 
field was obtained by solution of the linear balance 
equation 

V^ij; + Vi|^-Vf/f = V^4) /f . 

Equatorward of 2 5° north and south , the field was deter- 
mined using 

^ = <J>o/f 

where f is a mean coriolis parameter. Then a new geopoten- 
tial field (4)) was obtained by solution of the linear 
balance equation 

V^ = (fV^ip + Vi^-Vf) 

At 25° north and south, ^ and <t> were a combination of half 
the i> and (p from the poleward case and half the ip and (j) fr -m 
the equatorward case. It was hoped that this would reduce 
the amplitude of the high frequency oscillations in the 
tropics (Winninghoff , 1971). 



By initial geopotential field is meant either the FNWC 
objective analysis of <j) or a simulated <t> field-obtained by 
solution of Equation (h) using the analytic function form 
of ^ (Equation (1) ) . 



2k 



IV. WAVE ANALYSIS METHOD 
A Fourier series was determined at each five degrees of 
latitude around the latitude circle. The employment of such 
a series, known as harmonic analysis, is extensively used 
in the study of observational data (Jeffreys and Jeffreys, 
1956). With this technique, the phase angles and amplitudes 
of each wave number around a latitude circle can be calcu- 
lated. A Fourier series can be expressed as follows 
(Heburn 1972): 



F(x) = A^ + E (A cos mx + B sin mx) 

o ^ n^ m 

m 

= C + I (C cos (mx - 5m) 
o ^ ni 
m 



where 



B A 

c = m _ m 
m sin(6m) cos (6m) 



and 



1 B 
6^ = tan"\/^). 



The first three experiments involved an input stream function 
of wave number six. The values of primary interest, there- 
fore, were 6r and Cr which are the phase angle and amplitude 
of v;ave number six. Other phase angles and amplitudes were 
extracted and examined, especially in the real data case. 

25 



V. RESULTS 

Four experiments will be presented in this section. The 
first three of these experiments were performed with analytic 
fields. These analytic fields were derived from a stream 

function with a wave number of 6 and arbitrary constant, 

2 -1 
A, set equal to 1000 m sec . The vertical temperature 

structure of these fields was determined by the NACA stan- 
dard atm.osphere. Experiments I and II consisted of two 
36-hour forecasts each. The first forecast used "analytic" 
winds and the second forecast "derived" winds. Experiment 
III consisted of one 36-hour forecast using "analytic" 
winds. Experiment IV was performed with FNWC analyses and 
consisted of one 36-hour forecast using "derived" winds. 
Experiment I. The analytic geopotential field used in this 
experiment was derived from a stream function with a wave 
number of 6 and phase speed of -13° longitude per day. T .e 
initial surface pressure analysis is shown on chart E. Tie 
forecast surface pressure fields using both "analytic" and 
"derived" winds can be found on charts F-H. 
Experiment II . The analytic geopotential field used in 
this experiment was derived from a stream function with a 
wave number of 6 and phase speed of 0° longitude per day. 
The initial surface pressure analysis is shown on chart I. 
The forecast surface pressure fields using both "analytic" 
and "derived" winds can be found on charts J-L. 



26 



Table I compares the wave amplitudes and mean heights 
of the 36-hour forecast pressure field for experiment II 
(analytic winds) with those of the initial pressure field 
at selected latitudes. The amplitudes and mean heights in 
this experiment decreased at latitudes equatorward of 30° 
north (south) and increased poleward of 30° north (south). 
These amplitude variations are believed to be caused by 
nonlinear effects. 

In addition, Figures 6-8 are plots of terrain pressure 
versus forecast hour at selected latitudes. These terrain 
pressures are plotted every hour out to 36 hours. They 
show small amplitude internal gravity waves with periods 
ranging from 6-12 hours. These oscillations are believed 
to be due to the geostrophic adjustment mechanism and are 
peculiar to the type of analytic field used. 
Experiment III . The analytic field used in this experime* t 
was derived from a stream function with a wave number of ' 
and a phase speed of 12° longitude per day. The initial 
surface pressure analysis is shown on chart M. The fore- 
cast surface pressure fields using "analytic" winds only 
can be found on charts N-P. 

Experiment IV. This experiment was performed using FNWC 
objective analyses for OOZ February 8, 1973. The initial 
surface pressure analysis is shown on chart A. The surface 
pressure analysis for 12Z February 8, 1973 is shown on 
chart B. The 12-hour forecast made by the current F^^WC 



27 



model is shown on chart C. The 12-hour forecast made by 
the global m.odel is shown on chart D. 

It should be noted that Experiment IV included terrain, 
frictional stress, heating and moisture effects as discussed 
in Kesel and Winninghoff (1972). 

Table II summarizes the results of all experiments 
performed. The analytic phase speeds are compared to the 
actual phase speeds for all experiments using analytic data. 
The table also includes A which is proportional to the 
amplitude of the analytic wave, B which is a function of 
the phase speed and wave number, the method of balancing 
and the time required to make a 36-hour forecast. 

The forecast fields in all the experiments using analytic 
data showed a considerable tilt backward at high latitudes 
in the phase propagation of the wave. This was to be ex- 
pected since the Arakawa averaging technique tends to smcth 
the gradients at high latitudes. In addition, the Arakaw . 
technique gives an effective Ax which is comparable to tY it 
at low latitudes, thus as the wavelength decreased toward 
the poles the phase speed also decreased (Gates, 1959). 

The result of this differential movement, which was 
more pronounced in the "derived" wind cases, was the forma- 
tion of closed highs and lows at the higher latitudes which 
propagated equatorward. This distortion was aggravated by 
nonlinear effects introduced after the field ceased to be 
harmonic in the longitudinal direction. 



28 



Furthermore, in all the experiments using the "derived" 
wind, high frequency inert ial gravity vraves were generated 
due to the initial im.balance between the mass and wind 
fields in the tropics. These gravity waves caused a rapid 
deterioration of the harmonic six-wave pattern and added 
appreciably to the problem at the poles. The shear which 
developed at 25'^ north (south) was attributed to the method 
used to blend the v/inds between the tropics and mid-latitudes. 
The larger error in the phase speeds calculated for the 
"derived" wind cases were also attributed to this initialization 

Since these experim.ents were performed using a multi- 
level primitive equation model which allows divergence. 
Equation (3) was satisfied only approximately. Also, 
Rossby (1939) has shown that the presence of divergence in 
a barotropic atmosphere will slow the rate of wave propaga- 
tion, especially for small values of wave number. It was 
not surprising, therefore, that the actual phase speed Wc.: 
always less than expected. 

Finally, a thorough examination was made of Experiment 
IV to determine the general quality of the program, that 
is, if the inertial-gravity motions were being controlled 
realistically so that the model produces consistent meteor- 
ological appearing results. Figures 3-5 give the surface 
pressure oscillations at each hour of a 36-hour forecast 
for selected latitudes using the full global model. In 
addition. Table III compares forecast pressures of the 
global model to both the verifying analysis and forecasts 



29 



made by the FNV/C model for selected lows and highs. In 
interpreting these results one should keep in mind the 
crude initialization. Also, for those not accustomed to 
seeing values of terrain pressure at each hour in a primitive 
equation model, these oscillations, although larger than 
normal at these points, are by no means unusual for such a 
poorly initialized integration. There is no artificial 
smoothing done and recall that the run begins with a linear 
balance wind. In addition, there are none of the various 
devices which operational experience has shown is necessary 
to give an acceptable short term product, i.e., not calling 
on convective adjustment for the first several time steps. 
Nevertheless, these results are stable and retain the 
dominant meteorological scale without difficulty. 



30 



TABLE I 



Latitudes 



Amplitude of Wave #6 (Cg) 



Mean Pressure (C-, ) 



Equator 
10° N 
20° N 
30° N 
^0° N 
50° N 
60° N 
70° N 
80° N 



Initial 

Pressure 

Field 


. 5. 


.4 


18. 


.2 


29. 


.1 


30. 


.2 


21. 


.5 


10. 


.1 


2. 


.7 


1 


.31 




.01 



36-Hour Forecast Initial 36-Kour Forecast 
Pressure Pressure Pressure 

Field 

11.0 
24.0 
30.2 

25.5 
15.0 

5.8 

1.5 
.37 



Field 


Field 


1051 


1040 


1035 


1034 


1016 


1015 


995 


995 


973 


977 


953 


956 


935 


937 


922 


924 


914 


919 



Note: Data is from Experiment II (Analytic Winds) 



TABLE I. Wave Amplitude and Mean Heights for Selected 
Latitudes and Forecast Time. 



31 



'^' 



o 

in 



o 
o 



IS 

o 
o 

CO 



s 




CQ 




rH W (U O 
CD Cd <U iH 



^S 






CO 


CO 




CVi 


rH 


C\J 


cr\ 


H 


1 


1 


1 


1 



OJ 



I 



CO 

I 



CO 

o 



vo 



o 
o 
o 



CO 

1 



in 



I 



CO 

+ 



o 



VD 



o 
o 
o 






I 



in 
o 



o 




o 


•H 


•d 


•H 


4-> 


(1) 


+3 


>s W 


> CO 


>5 CO 


rH T? 


•H 'O 


r^ 73 


Cd C 


^ c 


Cd c 


C -H 


0) ^H 


C -H 


< rs 


q:s 


<:s 



vo 



m 
i 



CO 



vo 
o 



•d 

<u 

> CO 
•H 73 
U C 
Q) "H 
Q ^ 



VO 



t-- 


VO 

1 


C— 


VO 

1 


• 


X o 


• 


X o 


CM 


rH 


rvj 


iH 


O 




o 




O 




o 




O 




o 




rH 




t-i 













4J 










c 




B 


O 

•H 




(1> 

6 




o 


-P 








CO 




•H 


O 


o 


O 




;^ 




o 


T-l w 




<D 




in 


M (U 


• 


CI, 
X 

w 




vo 


Cd nH 


o 




cd 


y <D 


-p 

0) 






CO 




Cm 




S H . 


•\ 


o x: 




o 


rH ^ 


CO 


o 
Td cd 


t~ 


CO 


cd <D 
•H 


o 


+ 


d) 


CO J-i 


•H 


O CD 




4^ 


(D P 


4^ 


x: 




:3 T? o 


Cd 


-P fn 


rH 


t3 


inclu 

as F 

erpol 


0) O 
S "M 

£ +3 


rH 


cu 


^ -p 


4-> CO 


+ 


5 


+J o 

to 

• 


5 
»» 


•H Cd 

0) 






fn CO 


CO 


bO S^H 




m 


(D 0) 


•H 


C O 


+ 


•H 


omput 
out in 


CO 

>3 


O fin 
rH 
< U 




^ 


o u 


CD 


CO O 




Hf 






^3 1 

:3 vo 

4J CO 

•H 

+j cd 
cd 


in 




in 




kJ 


o 




CO 




^ 


r-i 




rH 




ected 

to Ma 


O 








rH 


•H 




•CJ 




d) "TZi 


+5 




0) 




zn <D 


>J CO 




> CO 


u 


iH 'O 




•H Td 


^ -H 


Cd c 




U C 


' :3 


C -H 




<U -H 


D^ 


< rs 




Q :s 


■ (U 










^^ QC 










- <U 










.6 


vo 




1 




; -H 










Eh 










J 


vo 






CO Td 


Cvj 1 


1 


1 




cd c 


• X 


o 


1 




x: cd 


in 


rH 






bO 
13 C 


o 




1 




<U -H 


o 






4^ O 


o 




i 




:3 c 


iH 








Comp 
Bala 



H 



H 



M 



H 



OJ 
H 

+ 



M 
H 
M 



> 
M 



H 



EH 



32 



TABLE III 



Location of 
Pressure System 
on Initial 
Surface Analysis 

65°N 135°W 

70°N 15°E 

65°N 25°W 

65°N 70°W 

50°N 165°E 

55°N 115°E 

55°N 45°E 

50°N 55°W 

50°N 100°W 

50°N 160°W 

35°N 175°W 

30°N 20°W 

35°N 55°W 



Initial Surface Verifying 
Pressure Analysis Surface Pressure 
OOZ 8 Feb 1973 Analysis 

12Z 8 Feb 1973 



1042(H) 

971(L) 

991(L) 

999(L) 

966(L) 

10il9(H) 

.97ML) 

1031(H) 

1037(H) 

1004(L) 

1030(H) 

1006(L) 

1012(L) 



10^1 

975 
990 
992 
962 

1046 
976 
1030 
1037 
1009 
1030 
1009 
1006 



FNWC 


Global 


12-hour 


12 -hour 


Surface 


Surface 


Pressure 


Pressure 


Analysis 


Analysis 


1040 


1042 


976 


973 


989 


989 


992 


992 


959 


962 


1043 


1052 


975 


975 


1029 


1036 


1037 


1040 


1004 


1005 


1029 


1031 


1007 


1003 


1007 


1010 



TABLE III. Comparison of 12-Hour Forecast Pressures of 
the FNWC Model and the Global Model Against 
the Verifying Pressure Analysis (Experiment IV) 



33 




w 
o 

<L) 
U 
O 

O 

O 



O 

o 
o 

W 
o 

LTv 



4^ 

cd 

o 
-p 

CO 

a 
o 

^ 
o 

w 

13 
CO 
fn 
<U 
> 

CO 
JO 
Q) 

u 
c 

•H 

rt > 

^H M 

<D -(J 



o 






u 

^ 0) 
O Cu 
rH X 



en 



faO 
•H 
(in 



O 

OJ 

o 



in 

r-{ 

o 



o 

O 



O 
O 



on 
o 
o 



aangsaaj ufeaasj, 



34 




o 
<^ 

2: 

o 
o 

w 

o 



4^ 

cd 

o 

w 
(0 
o 

u 
o 

m 

:3 

(0 
f^ 

> 

CO 
CO 
<D 
U 



cti 

U 
0) 
Eh 



O 

O 



> 



4J 
C 

6 
•H 

0) 



bO 
•H 
fin 



O 



LTv 



o 



in 
m 

CTv 



sanssaaj uT^adaj, 



35 






u 




o 




(m 




s 




o . 




in 




CO 




W 




o 




LP\ 




iH . 




-P 




CCJ 




^ 




:3 




o 




ffi 




-p 




w 




a 




o 




<D 




?H 




O 


-p 


fo 


w 




cd 


CO 


o 


:3 


<D 


to 


U 


^ 


O 


<u 


ptH 


> 


Cm 


<D 


O 


Jh 




:3 


U 


CO 


:3 


CO 


o 


(U 


w 






-■* 




• '4 • 




-:> 




. - M 




) +^ 




H C 




<U 




<M e 




O -H 




Jh 




+^ <U 




O ft 




iH X 




P^ H 



in 

faO 
•H 



o 
o 



in 
o 
o 



o 
o 
o 



in 



o 



in 



aanssaaj ufeaaaj, 



36 




m 




O 

o 
o 

W 
o 


o 




in 


m 




rH 

Cd 
o 


LTv 






C\J 




-p 
w 
cd 
o 

(D 




4J 


O 




W 


Ph 




(ti 




o 


O 


w . 


C\J 


CD 


13^ 




^ 


CO CO 




O 


U ^ 




fc 


CD C 

> ^ 




<;h 


:s 




o 


cu 




^H 


^ -H 


in 


:3 


CO 4^ 


<H 


o 


CO >s 




K 


rain Pre 
II (Anal 


o 




f^ 


iH 




(D 4-- 

Eh C 

CU 

<M 6 

O -H 
U 

O D. 
iH X 

P-i W 



0) 

hO 

•H 



O 

vo 
o 



o 
in 
o 



o 
o 

iH 



O 

o 

rH 



o 

OJ 

o 



aj:nss8J.<j uif^aaaj, 



37 




CO 

ti 
o 

0) 

u 
o 

Ph 
<M 

o 

u 
o 



o 

<M 

o 
O 

.=r 

W 

o 

in 



d 
U 
o 

■P 

CO 

cd 
o 

(D 
U 
O 

w 

> 



u o 

13 -H 

CO 4-5 
CO >s 
0) rH 

U cd 

< 

•H 

cd M 
?^ M 
U 

d) 4J 
H C 
0) 

e 



o 



0^ w 



o 



hO 
•H 



o 

CVI 

o 



o 
o 



o 
o 
o 



O 



a<inss8JcI u"feaaaj, 



38 




u 
o 

Cm 
O 

in 

CO 

o 

un 

iH 

cd 

o 

-p 

CO 

cd 
o 

<D 
U 
O 

w 

:=s^ 
w w 

0) C 

> nH 



o 



0) 

to 4J 
<U rH 

u a 

CM c 

< 



cd H 

^ M 
EH 



O 



•4-3 

c 



p. 

iH X 



4^ 
O 



CO 

bO 
•H 



ON 



o 



O 
CO 



O 

CVJ 



o cr\ 

rH O 
0\ CT\ 



aanssaaj ufBadaj, 



39 




f mi m FEB 73-.ps. 



4^95SFC\flNflL V/iGtlJBPL MGDlL 



tt' lOtBIC* rffl'^F CEhTffPt u. 5. I*'* 



CHART A. Initial Surface Pressure Analysis 
(Experiment IV) 



40 




•-.^ 




-v^. ■ ^ "^ '^ ^^^^^^ .--- 


\s«» 


4- "/■■ 




<_ 




■'■•■■""■■\ 







12Z 08 FEB 75-FS/ 


4760 REPORTS 


■,1.5377.3' 


'■"^ 5BG 






"■■- ..J_ /■' \ -■"" 


^- ' 


aci' luwia*. «c«'i<» ctn'iio. u. s. ww.^ 1 



CHART B. Verifying Surface Pressure Analysis 
(Experiment IV) 



^1 




CHART C. 12-Hour Forecast Using FNWC Model 
(Experiment IV) 



42 







g 00Z 08 FEB 75-.P.S 



\.p HOlilR PROG y/^CLiD^pL'^MODEL 



I FLU1 itJfWlOl tCiiTO ttH'W*, u. 5. t«»ir» 



CHART D. 12-Hour Forecast Using Global Model 
(Experiment IV) 



43 




RNflLYTlC PSrC'flNfiL V,:£l:OBPL MODEL 



ti.',»c»tMC* rfai<^ ahiwi u. s- (*><< 



CHART E. Initial Surface Pressure Analysis 
(Experiment I) 



l\l\ 




1 2 HOUR FROG V./Gt:1Jj3flL 110DEL 



fltE' MJtltaiCflt UtS'rttR CENTPiH. U. S. N6 



Derived 
Winds 



Analytic 
Winds 







j2 HOUR PROG v;/GLljppC "MODEL 



fUll HJ^f>K« MCfllf^Jl aNlRfll U. S. hP'.l 



CHART P. 



12-Hour Surface Pressure Forecast Using 
Analytic and Derived Winds (Experiment I) 



45 




Analytic 
Winds 



H HOUR PROG V/CLOBaL- MODEL 



fLttl MjftRICflL UtMTHCR aSTFSl U. S. JlHV 



Derived 
Winds 




in HOUR PROG V/.'-GtOBPL -flODPL 



flt£' HUftflCfil g£flI»tR aWWl U. S. Nl!»» 



CHART G. 24-Hour Surface Pressure Forecast Using 

Analytic and Derived Winds (Experiment I) 



46 




35 HOLiFi PROG y/:-GLliUHL MOULL 



niL^ Wjttfli.Hl ^H"h».« (.iMt-M', t.'. S. ffl'i 






Derived 
Winds 



A ' * '"' .V- ' "-' . . 



Analytic 
Winds 




I 36 HOUR PROG V'/XLODfiL MODEL 



1 fLtE.1 NurtPICK itfP'HtB CtNTPi^L U. S. W=V' 



CHART H. 36-Hour Surface Pressure Forecast Using 

Analytic and Derived Winds (Experiment I) 



^7 




CHART I. Initial Surface Pressure Analysis 
(Experiment II) 



48 




Analytic 
Winds 



12 HOUR PROG v'/.'-GLOf.iHL fiuDLL 



fltf-I HiUKf- l>:ni. UCMiHLh CC'j'finL u. s. 



Derived 
Winds 




L._. 



; i: HOUR FROG '•/..•-'■Gl-OieHL MODE L 



fiU' iMtniCn uts'BEs ammn u. s. m>ji 



CHART J. 



12-Hour Surface Pressure Forecast Using 
Analytic and Derived Winds (Experiment II) 



^9 



Derived 
Winds 



Analytic 
Winds 




CHART K. 



24-Hour Surface Pressure Forecast Using 
Analytic and Derived Winds (Experiment II) 



50 



Derived 
Winds 



Analytic 
Winds 




i 36 HOUR FROG V.{i:iLl3BPL MODtL 



CHART L. 36-Hour Surface Pressure Forecast Using 

Analytic and Derived Winds (Experiment (II) 



51 




mi 17 RPR A-g-.^s, 



PNflLYTIC, PSFCinNRL V./Gtt33PL 'rODEL 



^Lttl <fcrti<lCA. i£fil<F XIi-WL u. 3. i*Y^ 



CHART M. Initial Surface Pressure Analysis 
(Experiment III) 



52 




5 nCi.--; PROG V./^cijBf^L -f^ODEL 



FLtET MJif.CPL .tfllMtS CDllRfv U. 5. N0»», 



Analytic 
Winds 



Analytic 
Winds 




i )2 HOl^R PROG y/Strdppt '"MODEL 



FlUT NUf^lCn. t£Pl»CJI CZMliWl U. S. r«)W, 



CHART N. 



6-Hour and 12-Hour Surface Pressure Forecast 
Using Analytic Winds (Experiment III) 



53 



Analytic 
V/lnds 




Analytic 
Winds 



A HOUR PROG Y.^fictj^PL "MODEL 



fLEii pumica KsiicB cdhrr. u. s. ftf. 



CHART 0. l8-Hour and 2^-Hour Surface Pressure Forecast 
Using Analytic Winds (Experiment III) 



5^ 




't i:>.f-^ ■ 

32 nUc^R '^ROG y./£t:Op-fiL '•■MODEL 



Analytic 
Winds 



Analytic 
Winds 




\ 36 HOQJR PROG y/£Lti^pPMODEL 



,-^\ 



ftUT lutjiica. ktsKifn :ixim. u. s. •»♦, 



CHART P. 30-Hour and 36-Hour Surface Pressure Forecast 
Using Analytic V/inds (Experirnent III) 



55 



VI. CONCLUSIONS 

Three cases of analytic data and one case of real data 
were numerically integrated using a 5-level baroclinic 
primitive equation model of the general circulation. 
Experiments v/ere performed using winds derived from the 
linear balance equation and winds derived analytically. 
The feasibility of using the linear balance equation to 
initialize the wind field was examined. In all cases, the 
forecasts remained meteorological and reasonably well-behaved. 
The forecasts using winds analytically derived were virtually 
free of small scale inertial-gravity motions, while the 
forecasts using winds from the linear balance equation 
excited intertial-gravity waves which were large and unde- 
sirable for operational forecasts. 

Therefore, an important question to be answered is wl it 
method of balancing should be used to initialize the glol al 
model so that it will not suffer during the early part o 
the forecast run from excitation of excessive inertial 
gravity motions. A number of solutions have been proposed 
by Myakoda and Moyer (1968), Nitta and Hovermale (1969), 
and Winninghoff (1971). 

The method examined by Winninghoff used the equations 
in the model itself in an iterative sense either at a fixed 
time level or even in a four dimensional sense in which 
data is assimilated into a running model. This allows for 



56 



the natural adjustment mechanism Itself to achieve the 
desired balance. A time scheme such as the Euler-backward 
may be used for selective damping of high frequency waves. 
The iterative method proposed by Winninghoff has the advan- 
tage of mathematical simplicity and complete consistency 
with the prediction model. Unfortunately, Winninghoff 
(1971) estim.ates that the iterative procedure must continue 
for an equivalent of an I8 to 2k hour forecast. This, of 
course, is not operationally feasible at the present time. 
Consequently, the technique which is now being tested by 
PNV/C utilizes the best vertical mass structure possible obtained 
by the latest variational analysis techniques available 
and then solves the complete balance equation at each level 
in order to get the initial wind fields. This initialization 
along v/ith other helpful operationally tested devices such 
as smoothing, turning friction on slowly and not calling 
on convective adjustment for the first several time steps 
should be a satis-factory interim solution to this most 
difficult problem. 



57 



LIST OF REFERENCES 



1. Arakawa, Akio, Katayama, Akira, and Mintz, "Numerical 
Simulation of the General Circulation of the Atmosphere," 
Proceedings of the V/MO/IUGG Symposium on Numerical 
Weather Prediction , Tokyo, Japan, November 25 - December 
W] 196b, Japan Meteorological Agency, Tokyo, Mar. I969, 
pp. IV-1 - IV-14. 

2. Arakawa, Akio, "Computational Design for Long-Term 
Numerical Integration of the Equations of Fluid Motion: 
Two Dimiensional Incompressible Flow. Part I," Journal 
of Computational Physics , Vol. 1, No. 1, Academic 
Press, Inc., New York, N.Y., Jan. I966, pp. 119-1^3. 

3. Burington and Torrance, Higher Mathematics , p. ^2^- 
429, McGraw-Hill, 1939. 

H. Dickson, Robert R. , and Posey, Julian, "Maps of Snow- 
Cover Probability for the Northern Hemisphere," 
Monthly Weather Review , Vol. 95, No. 6, June I965, 
p. 347-353. 

5. Gates, W. L. , "On the Truncation Error, Stability and 
Convergence of Difference Solutions of the Barotropic 
Vorticity Equation," J. Meteor. , V. I6, p. 556-568, 
1959. 

6. Gates, W. L. , and Riegel, C. A., "A Study of Numerical 
Errors in the Integration of Barotropic Flow on a 
Spherical Grid," J. of Geoph. Res. , V. 67, No. 2, 

p. 773-784, Feb. 19^^: 

7. Haltiner, G. J., Numerical Weather Prediction , p. 1-39, 
90-114, 193-196 and 220-243, Wiley, 1971. 

8. Haltiner, G. J., and Martin, F. L. , Dynamical and 
Physical Meteorology , p. 52-53, McGraw-Hill, 1957. 

9. Haurwitz, B. , 1940: The Motion of Atmospheric Distur- 
bances. J. Marine Research (Sears Foundation) , 

V. 3, p. 35-50. 

10. Haurwitz, B. , 1940: The Motion of Atmospheric Distur- 
bances on the Spherical Earth. J. Marine Research 
(Sears Foundation) , V. 3, p. 254-267. 



58 



11. Heburn, G. W. , Numerical Experiments with Several 
Time Differencing Schemes with a Barotropic Primitive 
Equation Model on a Spherical Grid , M. S. Thesis, 
Naval Postgraduate School, 1972. 

12. Jeffreys and Jeffreys, Methods of Mathematical Physics , 
p. ^29-^31, Cambridge, 1956. 

13. Kesel and Winninghoff, "The Fleet Numerical Weather 
Central Operational Primitive-Equation Model," 
Mon. Wea. Rev. , V. 100, No. 5, p. 360-373, 1972. 

14. Kreyszig, E. , Advanced Engineering Mathematics , 
p. 175-178, Wiley, 1962. 

15. Moyer, R. W. and Miyokoda, K. , "A Method of Initiali- 
zation for Dynamical Weather Prediction," Tellus , 

V. 20, p. 115-128, 1968. 

16. Naval Postgraduate School Report NPS-51Wu7108lA, 
Restorative-Iterative Initialization for a Global 
Prediction Model , by F. J. V/inninghof f , September 1971.- 

17. Neam.tan, S. M. , "The Motion of Harmonic Waves in the 
Atmosphere," J. Meteorology , V. 3, p. 53-56, 19^6. 

18. Nitta, T. and Hovermale, J. B. , "A Technique of 
Objective Analysis and Initialization for the Primitive 
Forecast Equations," Mon. Wea. Rev. , V. 97, p. 652- 
658, 1969. 

19. Philips, N. A., "The General Circulation of the 
Atmosphere: a Numerical Experiment," Quart. J. 
Meteor. Soc. , V. 82, p. 123-16^, 1956. 

20. Rossby, C. G. et al., "Relations Between Variations 
in the Intensity of the Zonal Circulation of the 
Atmosphere and the Displacements of the Semi- 
Permanent Centers of Action," J. Marine Res. (Sears 
Foundation) , V. 2, p. 38-55, 1939. 

21. UCLA Department of Meteorology Technical Report No. 3, 
Description of the Mint z-Arakawa Numerical General 
Circulation Model , by W. E. Langlois and C. W. .Kwok, 
1959^: 

22. Smagorinsky et al. , "Numerical Results From a Nine-Level 
General Circulation Model of the Atm.osphere , " Monthly 
Weather Review, Vol. 93, No. 12, Dec. 1965, PP- 727-768. 



59 



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Monterey, California 939^0 

6. Officer in Charge 1 
Environmental Prediction Research Facility 
Naval Postgraduate School 

Monterey, California 939^0 

7. Commanding Officer 1 
Fleet Numerical V/eather Central 

Naval Postgraduate School 
Monterey, California 939^0 

8. LCDR Wayne R. Lambertson 1 
Fleet Numerical Weather Central 

Naval Postgraduate School 
Monterey, California 939^0 

9. ARCRL - Research Library 1 
L. G. Hanscom Field 

Attn: Nancy Davis/Stop 29 
Bedford, Massachusetts 01730 

10. Director, Naval Research Laboratory 1 

Attn: Tech. Services Info. Officer 
Washington, D. C. 20390 



60 



11. American Meteorological Society 
^5 Beacon Street 

Boston, Massachusetts 02128 

12. Department of Meteorology 
Code 51 

Naval Postgraduate School 
Monterey, California 939^0 

13. Department of Oceanography 
Code 58 

Naval Postgraduate School 
Monterey, California 939^0 

14. Office of Naval Research 
Department of the Navy 
Washington, D. C. 20360 

15. Comm.ander, Air Weather Service 
Military Airlift Command 

U.S. Air Force 

Scott Air Force Base, Illinois 62226 

16. Atmospheric Sciences Library 
National Oceanographic Atmospheric 

Administration 
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17. National Center for Atmospheric Research 
Box 1470 

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18. Dr. T. N. Krishnamurti 
Department of Meteorology 
Florida State University 
Tallahassee, Florida 32306 

19. Dr. Fred Shuman 
Director 

National Meteorological Center 
Environmental Science Services 

Administration 
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20. Dr. J. Smagorinsky 
Director 

Geophysical Fluid Dynamics Laboratory 
Princeton University 
Princeton, New Jersey 085^0 

21. Dr. A. Arakawa 
Department of Meteorology 
UCLA 

Los Angeles, California 90024 



61 



22. Professor N. A. Phillips 1 
5^-1^22 

M. I. T. 

Cambridge, Massachusetts 02139 

23. Dr. Russell Elsberry 1 
Department of Meteorology 

Naval Postgraduate School 
Monterey, California 939^0 

2h. Dr. Jerry D. Mahlman 1 

Geophysical Fluid Dynamics Laboratory 
Princeton University 
Princeton, New Jersey 085^0 

25. Dr. Robert L. Haney 1 
Department of Meteorology 

Naval Postgraduate School 
Monterey, California 939^0 

26. Dr. Ron L. Albertv 1 

National Severe Storm Laboratory 
I6l6 Halley Circle 
Norman, Oklahoma 

27. Dr. W. L. Gates ^ 1 
The RAND Corporation 

1700 Main Street 

Santa Monica, California 90H06 

28. Dr. Richard Alexander 1 
The Rand Corporation 

1700 Main Street 

Santa Monica, California 90406 

29. Commanding Officer 1 
Fleet Weather Central 

Box 110 

FPO San Francisco 966IO 

30. Dr. F. J. Winninghoff . 1 
Department of Meteorology 

UCLA 

Los Angeles, California 9002^ 

31. LCDR P. G. Kesel ' 1 
ODSI 

2^60 Garden Road 

Monterey, California 939^0 • 



62 



32. Mr. Leo C. Clarke 
PNWC 

Naval Postgraduate School 
Monterey, California 939^0 

33. Naval Weather Service Command 
Washington Navy Yard 
Washington, D. C. 20390 



63 



Sfcuritv Classification 



DOCUMENT CONTROL DATA -R&D 

iSfcurity f f.ss.fic.don o( title, body ot abstract and indexine annotation must be •nitred u-hen the overall ttpott Is cUsslllod) 



iN*TtfsiG ACTIVITY (Corporate author) 



Naval Postgraduate School 
Monterey, California 939^0 



2». REPORT SECURITY C L A SSI F I C A T I Oh 

Unclassified 



26. CROUP 



EPOR T TITLE 



Numerical Experiments with a Five-Level Global Atmospheric 
Prediction Model Using a Staggered, Spherical, Sigma Coordinate 
System ^ 



ESCRIPTIVE NOTES (Type of report and,lncluaive datet) 

Master's Thesis; March 1973 



U T HO R IS I ff"*' n«me, middle initial, latl name) 

William Theodore Elias 



EPORT DATE 

March 1973 



CONTRACT OR GRANT NO. 



PROJEC T NO. 



DISTRIBUTION STATEMENT 



T. TOTAL NO. OF PACES 



65 



7b. NO. OF REFS 



21 



Sa. ORIGINATOR'* REPORT NUMBERIS) 



86. OTHER REPORT NO(S) (Any other numbmre that may be aaalg^ed 
thie report) 



Approved for public release; distribution unlimited. 



SUPPLEMENTARY NOTES 



12. SPONSORING MILITARY ACTIVITY 

Naval Postgraduate School 
Monterey, California 939^0 



ABSTRACT 



Three cases of analytic data and one case of real data wer 

numerically integrated using a 5-level baroclinic primitive 

equations model of the general circulation. Experiments were 

performed using initial winds derived from the linear balance 

equation and also winds derived analytically. The feasibility of 

using the linear balance equation to initialize the wind field was 

examined. In all cases, the forecasts remained meteorological and 

reasonably well-behaved. Nevertheless, the forecasts derived from 

initial winds generated by the linear balance equation excited 

large, operationally-undesirable inertial-gravity waves, while 

the forecasts from analytically determined initial winds remained 
free of such small scale "noise". 



FORM 1/L7'3l (PAGE 1) 

I NOV *S t *^ / w 

N 0101 -807-681 1 



6i| 



Security Clacaification 



A-ai4oa 



Security Classifirntion 



KEY yvo ROI 



Numerical 
Atmospheric 
Global 
Baroclinic 
Staggered Grid 

Spherical Coordinates 
Sigma Coordinate System 
Primitive Equation 



ROLE WT 



DD ,Z\'.A473 'B*^'^: 



S/N OIOI-807.6S2I 



65 



Security Cl«B>ification 



A- 3M09 



Thesis 
:316 
c.l 



U2023 

Elias 

Numerical experiments 
with a five-level global 
atmospheric prediction 
model using a staggered, 
spherical, sigma coordi- 
nate system. 



'>?> NOV 73 



21683 



/ 


1 


1 


Thesis 
E316 
c.l 


11+2023 
Elias 

Numeric;^! experiments 
with a five-level global 
atmospheric prediction 
model using a staggered, 
spherical, sigma coordi- 
nate system. 




r^^. 







thesE316 

Numerical experiments with a five-level 



i 





3 2768 001 89278 9 

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